Economics Lecture 4 - Univerzita Karlovavitali/documentsCourses/Lecture 4...Economics Lecture 4 2016-17 Sebastiano Vitali Course Outline 1 Consumer theory and its applications 1.1

Economics Lecture 4

2016-17

Sebastiano Vitali

Course Outline

1 Consumer theory and its applications

1.1 Preferences and utility

1.2 Utility maximization and uncompensated demand

1.3 Expenditure minimization and compensated demand

1.4 Price changes and welfare

1.5 Labour supply, taxes and benefits

1.6 Saving and borrowing

2 Firms, costs and profit maximization

2.1 Firms and costs

2.2 Profit maximization and costs for a price taking firm

3. Industrial organization

3.1 Perfect competition and monopoly

3.2 Oligopoly and games

1.3 Expenditure

minimization and

compensated demand

1.3 Expenditure minimization and

compensated demand

1. Definitions of compensated & uncompensated demand

2. Definition of the expenditure function

3. Homogeneity of the compensated demand and

expenditure functions

4. Income & substitution effects

5. The slope of compensated demand curves

6. Compensated demand & the expenditure function with

Cobb-Douglas utility

Expenditure minimization and compensated

demand

7. Compensated demand & the expenditure function with

perfect complements and perfect substitutes utility

8. Properties of the expenditure function

9. The Slutsky equation

Expenditure minimization and compensated

demand

Why bother?

1. Income and substitution effects, essential for

understanding the effects of changes in wages and

taxes on labour supply and interest rates on savings.

2. Expenditure function, essential for consumer

surplus and welfare economics.

Definitions of

compensated and

uncompensated

demand

What we have been calling demand up to now is

uncompensated (Marshallian) demand which maximizes

utility u given prices p1 and p2 and income m, so is a

function of p1, p2, m,

notation x1(p1,p2,m), x2(p1,p2,m).

Compensated (Hicksian) demand minimizes the cost of

obtaining utility u at prices p1 and p2 and is a function of

utility u, p1, p2, notation h1(p1,p2,u), h2(p1,p2,u).

1. Definitions of compensated and

uncompensated demand

0 x1

x2

To get uncompensated demand fix

income and prices which fixes the

budget line.

Get onto highest possible

indifference curve.

To get compensated demand fix

utility and prices which fixes the

indifference curve and gradient of

budget line.

Get onto lowest possible budget

line.

x2

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

x2



budget line.


indifference curve.




budget line.


line.

x2

0 x1

0 x1

Definition of the

expenditure function

and intuition

The expenditure function E (p1, p2,u) is the minimum

amount of money you have to spend to get utility u with

prices p1 and p2. It is a function of p1, p2 and u.

The amount of goods which minimizes the cost of

getting utility u is compensated demand h1 (p1, p2,u),

h2 (p1, p2,u)

so E(p1, p2,u)= p1h1 (p1, p2,u) + p2 h2 (p1, p2,u).

2. Definition of the expenditure function

A student buys 100 packs of sandwiches a year. The sandwich

price rises from €1 to €1.50. Could the student maintain the

same level of utility with

€50 more?

€60 more?

€40 more?

€20 more?

Intuition for the expenditure function

© Getty Images




€50 more? yes

€60 more?

€40 more?

€20 more?


© Getty Images




€50 more? yes

€60 more? yes

€40 more?

€20 more?


© Getty Images




€50 more? yes

€60 more? yes

€40 more? ?

€20 more?


© Getty Images




€50 more? yes

€60 more? yes

€40 more? ?

€20 more? ??


© Getty Images

Given a choice between grapes, cherries and

apple you chose grapes.

Are you worse off if you are forced to chose

between grapes, apricots & blackberries?

No because you can still get grapes.

The logic of choice

Given a choice between grapes, cherries and

apple you chose grapes.

Are you worse off if you are forced to chose

between grapes, apricots & blackberries?

No because you can still get grapes.

The logic of choice

Logic of choice

• If €10 is the cost of the cheapest way of doing something

any other way of doing it costs €10 or more.

• If the range of things you can do changes, but you can

still do what you did before you are no worse off.

• If prices change but you get just enough extra money to

carry on doing the same thing you are no worse off.

Homogeneity of the

compensated demand

and expenditure functions

A function f(z1,z2,z3…..zn) is homogeneous of degree 0 if for all

numbers k > 0

f(kz1,kz2,kz3…..kzn) = k0 f(z1,z2,z3…..zn) = f(z1,z2,z3…..zn).

Multiplying z1,z2,…..zn by k > 0 does not change the value of f.

A function f(z1,z2,z3…..zn) is homogeneous of degree one if

for all numbers k > 0

f(kz1,kz2,kz3…..kzn) = k1 f(z1,z2,z3…..zn) = kf(z1,z2,z3…..zn)

Multiplying z1, z2 …zn by k multiplies the value of f by k.

Mathematical definition of homogeneous functions

3. Homogeneity of the compensated

demand and expenditure functions

Compensated demand is homogeneous of degree 0 in

prices.

If k > 0 h1(kp1,kp2,u) = h1(p1,p2,u).

Expenditure function is homogeneous of degree 1 in prices.

If k > 0 E(kp1,kp2,u) = k E(p1,p2,u).

The next slides explain why.




budget line.


line.

x2

0 x1

Compensated demand depends on the indifference curve

and the slope –p1/p2 of the budget line.

Multiplying p1 and p2 by k does not change the slope so

does not change compensated demand so

h1(p1,p2,u) = h1(kp1,kp2,u) h2(p1,p2,u) = h2(kp1,kp2,u).


prices.

From the definition of the expenditure function

E(p1, p2,u)= p1h1(p1, p2,u) + p2 h2(p1, p2,u)

&

E(kp1, kp2,u)= kp1h1(kp1, kp2,u) + kp2 h2(kp1, kp2,u)


E(p1, p2,u)= p1h1(p1, p2,u) + p2 h2(p1, p2,u)

&


Since compensated demand is homogeneous of degree 0

in prices:



E(p1, p2,u)= p1h1(p1, p2,u) + p2 h2(p1, p2,u)

&


Since compensated demand is homogeneous of degree 0

in prices:


E(kp1, kp2,u) = kp1h1 (p1, p2,u) + kp2 h2 (p1, p2,u) =

= k [p1h1 (p1, p2,u) + p2 h2 (p1, p2,u)] = k E(p1, p2,u)

The expenditure function is homogeneous of degree 1

in prices.

Income and substitution

effects and compensated

and uncompensated

demand

0 m/p1 x1

x2

Income and substitution effects

A

0 m/p1 x1

x2


A

The price of good 1

increases from p1 to p1’.

0 m/p1’ m/p1 x1

x2


A

The price of good 1


0 m/p1’ m/p1 x1

x2


AC

The price of good 1


0 m/p1’ m/p1 x1

x2


AC

The price of good 1


0 m/p1’ m/p1 x1

x2


AC

The price of good 1


0 m/p1’ m/p1 x1

x2


AC

The price of good 1


0 m/p1’ m/p1 x1

x2


AC

The price of good 1


0 m/p1’ m/p1 x1

x2


AC

The price of good 1


0 m/p1’ m/p1 x1

x2


A

The price of good 1


C

0 m/p1’ m/p1 x1

x2


A

B

The price of good 1


C

0 x1

x2


A

B

The price of good 1


Substitution effect A to B,

change in compensated

demand due to p1

C

0 x1

x2


A

B

The price of good 1




demand due to p1

Income effect B to C,

change in uncompensated

demand due m

C

0 x1

x2


A

B

The price of good 1




demand due to p1

Income effect B to C,

change in uncompensated

demand due m

Total effectC

The logic of

compensated

demand and the


• By definition

compensated demand h1(p1,p2,u), h2(p1,p2,u)

is the cheapest way of getting utility u at prices p1, p2

and costs p1 h1(p1,p2,u) + p2 h2(p1,p2,u) = E(p1,p2,u)

• Therefore any other way of getting utility u costs the

same or more at prices p1, p2

• Thus if u(x1,x2) = u then E(p1,p2,u) ≤ p1x1 + p2x2

0 x1

x2

(h1,h2)

(h1,h2) is by definition

the cheapest way of

getting utility u at

prices (p1, p2).

So at prices (p1, p2)

any other (x1, x2) with

utility u must lie on or

above the budget line

through (h1,h2) with

slope - p1/p2

slope

- p1/p2

0 x1

x2

(h1,h2)

(h1,h2) is by definition

the cheapest way of

getting utility u at

prices (p1, p2).

So at prices (p1, p2)

any other (x1, x2) with

utility u cannot lie

below the budget line

through (h1,h2) with

slope - p1/p2

slope

- p1/p2

Same information as in previous slide

Compensated demand

curves cannot slope

upwards:

geometric proof

The substitution effect of an increase in the price of a

good decreases or leaves unchanged the demand for

the good.

The compensated demand curve can never slope

upwards.

p1

0 x1

IMPORTANT

RESULT

The next

slides explain

5. The slope of compensated demand

curves

compensated

demand curve

h1A(p1A,p2,u) h2A(p1A,p2,u) compensated demand with

prices p1A,p2 and utility u.

Therefore u(h1A,h2A) = u.

h1B(p1B,p2,u) h2B(p1B,p2,u) compensated demand with

prices p1B,p2 and utility u.

Therefore u(h1B,h2B) = u.

Geometric proof that the compensated

demand curve cannot slope upwards

h1A(p1A,p2,u) h2A(p1A,p2,u) is the cheapest way of getting

utility u at prices p1A,p2

u(h1B,h2B) = u so h1B,h2B is another way of getting

utility u

therefore h1B,h2B cannot cost less than h1A,h2A at prices p1A,p2

h1B(p1B,p2,u) h2B (p1B,p2,u) is the cheapest way of getting

utility u at prices p1B,p2

u(h1A,h2A) = u so h1A,h2A is another way of getting

utility u

therefore h1B,h2B cannot cost more than h1A,h2A at prices

p1B,p2

p1Ah1A + p2h2A ≤ p1Ah1B + p2h2B because h1A is cheapest at p1A

p1Bh1B + p2h2B ≤ p1Bh1A + p2h2A because h1B is cheapest at p1B

Geometric proof that the compensated


0 x1

x2

(h1A,h2A)slope

- p1A/p2

h1B,h2B cannot cost less h1A,h2A at prices p1A,p2

so (h1B,h2B) cannot lie in the shaded area.

(h1B,h2B)

0 x1

x2

(h1A,h2A)slope

- p1A/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)slope

- p1A/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)slope

- p1A/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)slope

- p1A/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)

slope

- p1B/p2

h1B,h2B cannot cost more than h1A,h2A at prices p1B,p2


(h1B,h2B)

0 x1

x2

(h1A,h2A)

slope

- p1B/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)

slope

- p1B/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)

slope

- p1B/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)

slope

- p1B/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)

slope

- p1B/p2



(h1B,h2B)

0 x1

x2

(h1A,h2A)

slope

- p1B/p2

Here p1B > p1A

(h1B,h2B) cannot lie in either shaded area

(h1B,h2B) must lies in the white triangle, so h1B ≤ h1A.

slope

- p1A/p2

(h1B,h2B)

0 x1

x2

(h1A,h2A)

slope

- p1B/p2

Here p1B > p1A

(h1B,h2B) cannot lie in either shaded area

(h1B,h2B) must lies in the white triangle, so h1B ≤ h1A.

slope

- p1A/p2

(h1B,h2B)

The compensated

demand curve can never

slope upwards.

Compensated demand

curves cannot slope

upwards:

algebraic proof

h1A(p1A,p2,u) h2A(p1A,p2,u) is the cheapest way of getting

utility u at prices p1A,p2

u(h1B,h2B) = u so h1B,h2B is another way of getting

utility u

Therefore at prices p1A,p2 quantities h1A,h2A cannot cost more

than h1B,h2B

so in algebra p1Ah1A + p2h2A ≤ p1Ah1B + p2h2B

Remember the logic used here to derive the inequalities

Algebraic proof that the compensated


h1B(p1B,p2,u) h2B (p1B,p2,u) is the cheapest way of getting

utility u at prices p1B,p2

u(h1A,h2A) = u so h1A,h2A is another way of getting

utility u

Therefore at prices p1B,p2 quantities h1B,h2B cannot cost more

than h1A,h2A.

so in algebra p1Bh1B + p2h2B ≤ p1Bh1A + p2h2A



Remember the logic used here to derive the inequalities

Inequalities from the two previous slides

p1Ah1A + p2h2A ≤ p1Ah1B + p2h2B

p1Bh1B + p2h2B ≤ p1Bh1A + p2h2A

Add the inequalities to get

p1Ah1A + p2h2A + p1Bh1B + p2h2B

≤ p1Ah1B + p2h2B + p1Bh1A + p2h2A



Remember the logic used here and derive the inequalities

Everything that follows comes from simplifying this inequality.

Simplify the inequality from the previous slides

p1Ah1A + p2h2A + p1Bh1B + p2h2B

≤ p1Ah1B + p2h2B + p1Bh1A + p2h2A

Subtract p2h2A and p2h2B from both sides to get

p1Ah1A + p1Bh1B ≤ p1Ah1B + p1Bh1A

Remember the logic used here and derive the inequalities

Everything that follows comes from simplifying this inequality.

From the previous slide

p1Ah1A + p1Bh1B ≤ p1Ah1B + p1Bh1A

Rearrange to get 0 ≤ ( p1B - p1A ) ( h1A – h1B)

so if the price rises from p1A to p1B so p1B - p1A > 0

either h1A – h1B = 0 so compensated demand does not change

or h1A – h1B > 0 so compensated demand falls.

Compensated demand

and the expenditure

function with Cobb-

Douglas utility

6. Compensated demand and the expenditure

function with Cobb-Douglas utility

Step 1: What problem are you solving?

The problem is minimising expenditure p1x1 + p2x2 subject to

non-negativity constraints x1 ≥ 0 x2 ≥ 0

and the utility constraint u(x1,x2) = x12/5x2

3/5 ≥ u.

Step 2: What is the solution a function of?

Compensated demand is a function of prices & utility so is

h1(p1,p2,u) h2(p1,p2,u)

Finding compensated demand with Cobb-

Douglas utility

Step 3: Check for nonsatiation and convexity

We have already done this, both are satisfied.

Why does this matter?

See the next slide.

0 x1

x2

if there is a tangency

point such as A

where MRS = p1/p2

and utility is u

this is compensated

demand

because any cheaper

point such as B gives

lower utility.

A

B

utility u

With nonsatiation and convexity

Essential logic


Douglas utility

Tangency requires that MRS = p1

p2

here we have already found

Step 4: Use the tangency and utility

conditions

MRS =

1

2

52

2

52

1

53

2

53

1

2

1

3

2

5

35

2

x

x

xx

xx

x

u

x

u

//

//

x12/5 x2

3/5 = u

2x2 = p1

3x1 p2

utility condition

tangency condition

Step 4: Use the tangency and utility conditions


Douglas utility

because if x12/5 x2

3/5 > u there is a

cheaper way of getting utility u or more.

0 x1

x2

A

utility u


Douglas utility

Step 5: Draw a diagram based on the

tangency and utility conditions

2x2 = p1

3x1 p2


Douglas utility

Step 6: Remind yourself what you are

finding and what it depends on.

You are finding compensated demand h1 and h2

which are functions of p1, p2 and u.

Step 7: Write down the equations to be

solved.

The equations are andx12/5 x2

3/5 = u


Douglas utility

Step 8 solve the equations and write down

the solution as a function.

.0),,(,0),,(

.2

3),,(

3

2),,(

),,(),,(

2

3

3

2

212211

5/2

2

1212

5/3

1

2211

212211

5/2

2

12

5/3

1

21

upphupph

up

pupphu

p

pupph

upphupph

up

pxu

p

px

Note

demand dcompensate for

notation using

gives This

Uncompensated and compensated demand

with Cobb-Douglas utility

.2

3),,(

3

2),,(

demand dCompensate

5

3),,(

5

2),,(

demand tedUncompensa

5/2

2

1212

5/3

1

2211

2

212

1

211

up

pupphu

p

pupph

p

mmppx

p

mmppx

The expenditure function with Cobb-Douglas

utility

upp

up

ppu

p

pp

upphpupphpuppE

up

pupphu

p

pupph

5/25/3

5/3

2

5/2

1

5/2

2

12

5/3

1

21

2122211121

5/2

2

1212

5/3

1

2211

2

3

3

2

2

3

3

2

),,(),,(),,(

isfunction eexpenditur theso

2

3),,(

3

2),,(

is demand dCompensate

Check

Compensated demand is function of prices and utility.

It is homogeneous of degree 0 in prices.

The expenditure function is a function of prices and utility.

It is homogeneous of degree 1 in prices.

Income and

substitution effects with


TP

0 m/p1’ m/p1 x1

x2

Income and substitution effects with Cobb-

Douglas utility

AC

The price of good 1


Income and the price of

good 2 do not change.

Demand moves from A to

C.

0 m/p1’ m/p1 x1

x2


Douglas utility

A

B

C

Substitution effect

A to B,

Keep utility and p2

constant

Stay on the same

indifference curve but

move to a different

tangency point because

the price ratio changes.

0 m/p1’ m/p1 x1

x2


Douglas utility

AC

The price of good 1


Substitution effect

decreases

increases

Income effect

decreases

B

0 m/p1’ m/p1 x1

x2


Douglas utility

AC

The price of good 1


Substitution effect A to B

decreases

increases

Income effect

decreases

B

0 m/p1’ m/p1 x1

x2


Douglas utility

AC

The price of good 1



decreases x1

increases x2.

Income effect

decreases

B

0 m/p1’ m/p1 x1

x2


Douglas utility

AC

The price of good 1



decreases x1

increases x2.

Income effect B to C

decreases

B

0 m/p1’ m/p1 x1

x2


Douglas utility

AC

The price of good 1



decreases x1

increases x2.

Income effect B to C

decreases both x1 and x2.

B



.2

3),,(

3

2),,(

demand dCompensate

5

3),,(

5

2),,(


5/2

2

1212

5/3

1

2211

2

212

1

211

up

pupphu

p

pupph

p

mmppx

p

mmppx

.2

3),,(

3

2),,(

demand dCompensate

5

3),,(

5

2),,(


5/2

2

1212

5/3

1

2211

2

212

1

211

up

pupphu

p

pupph

p

mmppx

p

mmppx

income effect: change in m on uncompensated demand



.2

3),,(

3

2),,(

demand dCompensate

5

3),,(

5

2),,(


5/2

2

1212

5/3

1

2211

2

212

1

211

up

pupphu

p

pupph

p

mmppx

p

mmppx

income effect: change in m on uncompensated demand

substitution effect: change on p1 on compensated demand



Income and substitution effects with


1

1

1

effect on substituti

effect income

p

h

m

x



03

2

5

3effect on substituti

0 1

5

2effect income

5/8

1

5/3

2

1

1

1

1

up

p

p

h

pm

x

Compensated demand

and the expenditure

function with perfect

complements utility

wheels x1

frames

x2

x2 = ½ x1

u(x1,x2) = min( ½ x1,x2)

x1 bicycle wheels,

x2 bicycle frames

if x2 < ½ x1 increasing x1

does not change utility

if x2 > ½ x1 increasing x1

increases utility.

7. Compensated demand and the

expenditure function with perfect

complements utility

0 x1

x2

x2 = ½ x1

u(x1,x2) = min( ½ x1,x2) = u

min( ½ x1,x2) = u

Finding compensated demand with perfect

complements utility

0 x1

x2

x2 = ½ x1

u(x1,x2) = min( ½ x1,x2) = u

Expenditure minimization

implies that (x1,x2) lies

at the corner of the

indifference curves so

x2 = ½ x1

and gives utility u so

½ x1 = x2 = u.

min( ½ x1,x2) = u

Finding compensated demand with perfect

complements utility

h1(p1,p2,u) = 2u, h2(p1,p2,u) = u.


with perfect complements utility

uupphupph

pp

mmppx

pp

mmppx

),,(2u ),,(

2),,( ,

2

2),,(

212211

21

212

21

211

uncompensated demand

compensated demand

Income and

substitution effects with

perfect complements

utility


perfect complements utility

effect on substituti

effect income


perfect complements utility

0effect on substituti

0 2

2effect income

1

1

21

1

p

h

ppm

x

0 m/p1’ m/p1 x1

x2

x2 = ½ x1

A

C

h1(p1,p2,u) = 2u, h2(p1,p2,u) = u.

Income and substitution effects with perfect

complements utility

0 m/p1’ m/p1 x1

x2

x2 = ½ x1

A

C

h1(p1,p2,u) = 2u, h2(p1,p2,u) = u.

The price of good 1



complements utility

0 m/p1’ m/p1 x1

x2

x2 = ½ x1

A

C

h1(p1,p2,u) = 2u, h2(p1,p2,u) = u.

The price of good 1


How much is the

substitution effect?


complements utility

0 m/p1’ m/p1 x1

x2

x2 = ½ x1

A

C

h1(p1,p2,u) = 2u, h2(p1,p2,u) = u.

The price of good 1


How much is the



complements utility

0 m/p1’ m/p1 x1

x2

x2 = ½ x1

A

C

h1(p1,p2,u) = 2u, h2(p1,p2,u) = u.

The price of good 1


How much is the



complements utility

0 m/p1’ m/p1 x1

x2

x2 = ½ x1

A

C

h1(p1,p2,u) = 2u, h2(p1,p2,u) = u.

The price of good 1


How much is the



complements utility

0 m/p1’ m/p1 x1

x2

x2 = ½ x1

A

C

h1(p1,p2,u) = 2u, h2(p1,p2,u) = u.

The price of good 1


There is no substitution

effect.

The income effect

reduces demand for x1

and x2.


complements utility

The expenditure function with perfect

complements utility

uppupup

upphpupphpuppE

uupphuupph

)2(2

),,(),,(),,(

isfunction eexpenditur theso

),,( 2),,(

is demand dCompensate

2121

2122211121

212211

Properties of the


8. Properties of the Expenditure

Function

1. Increasing in utility

2. The expenditure function increases or does not change when a price increases.

3. Homogeneous of degree 1 in prices.

4. Shephard’s lemma )u,p,p(hp

)u,p,p(E211

1

21

The expenditure function E (p1, p2,u) is the minimum

amount of money you have to spend to get utility u with

prices p1 and p2. It is a function of p1, p2 and u.

The amount of goods which minimizes the cost of

getting utility u is compensated demand

h1 (p1, p2,u), h2 (p1, p2,u)

so E(p1, p2,u)= p1h1 (p1, p2,u) + p2 h2 (p1, p2,u).

Definition of the expenditure function

0 x1

1. The expenditure function is increasing in

utility

u1

u2

E(p1,p2,u1)

p2

E(p1,p2,u2)

p2

x2

Utility increases

from u1 to u2.

The expenditure

function increases.

u1

u2

0 x1

x2

2. The expenditure function increases or

does not change when prices increase

A

B

The price of good 1

increases from p1A

to p1B,

compensated

demand moves

from A to B.

The expenditure

function increases.

The expenditure

function does not

change if demand

for good 1 is 0 at A.

E(p1A ,p2,u1)

p2

E(p1B ,p2,u2)

p2

3: The expenditure function is homogeneous

of degree 1 in prices.


prices.

If k > 0 h1(kp1,kp2,u) = h1(p1,p2,u).

Expenditure function is homogeneous of degree 1 in prices.

If k > 0 E(kp1,kp2,u) = k E(p1,p2,u).

Already explained.

Properties of the

expenditure function 4

Shephard’s Lemma

)u,p,p(hp

)u,p,p(E211

1

21

4. Shephard’s Lemma

)u,p,p(hp

)u,p,p(E211

1

21

p1

E(p1,p2,u)

gradient h1(p1A,p2,u)

0 p1A

derivative of expenditure = compensated

function w.r.t. p1 demand for good 1

The next

slides

explain

this.

u constant

p2 constant

p1 varies

• By definition

compensated demand h1(p1,p2,u), h2(p1,p2,u)

is the cheapest way of getting utility u at prices p1, p2

and costs p1 h1(p1,p2,u) + p2 h2(p1,p2,u) = E(p1,p2,u)

• Therefore any other way of getting utility u costs the

same or more at prices p1, p2

• Thus if u(x1,x2) = u then E(p1,p2,u) ≤ p1x1 + p2x2

h1A(p1A,p2,u) h2A(p1A,p2,u) compensated demand with

prices p1A,p2 and utility u.

Therefore u(h1A,h2A) = u and

p1Ah1A + p2 h2A = E(p1A,p2,u)

h1(p1,p2,u) h2(p1,p2,u) compensated demand with

prices p1, p2 and utility u.

Therefore u(h1,h2) = u

and p1h1 (p1,p2,u) + p2 h2(p1,p2,u) = E(p1,p2,u)

E(p1,p2,u) is the cheapest way of getting utility u at prices

(p1,p2)

(h1A,h2A) is another way of getting utility u, and

at prices (p1,p2) costs p1h1A + p2 h2A so

E(p1,p2,u) ≤ p1h1A + p2 h2A for all p1

E(p1A,p2,u) = p1Ah1A + p2 h2A

p1 price0 p1A

amount

spent

€

Cost of buying (h1A,h2A) at

prices (p1, p2) is p1h1A + p2 h2A

p1h1A + p2 h2A

is a straight line

with gradient h1A

New diagram

In this diagram (h1A,h2A) u and p2 do not vary, p1 varies.

E(p1,p2,u) ≤ p1h1A + p2 h2A for all p1

E(p1A,p2,u) = p1Ah1A + p2 h2A.

The graph of E(p1,p2,u) cannot be anywhere inside

the shaded area.

p1 price good 10 p1A

€ p1h1A + p2 h2A slope h1A

E(p1A,p2,u)

E(p1,p2,u)

p1 price0 p1A

€

E(p1A,p2,u)

The graph of E(p1,p2,u) meets the line with slope

h1A at A and is never above the line.

so the line is tangent to the graph of E(p1,p2,u) at A

so the derivative of E(p1,p2,u) with respect to p1 at A is h1A

(compensated demand for good 1)

p1h1A + p2 h2A slope h1A

)u,p,p(hp

)u,p,p(E211

1

21

The derivative of the expenditure function

E(p1,p2,u) with respect to p1

is compensated demand for good 1

Try with previous examples

Shepard’s Lemma

The Slutsky equation

),,(),,(),,(

),,(

),,(When

211211

1

211

1

211

21

mppxm

mppx

p

upph

p

mppx

uppEm

total effect of change in price

on demand at constant

income

substitution effect

of change in price

on demand, at

constant utility

income effect of

change in

income on

demand

-

9. The Slutsky equation

2TP

Why bother with the Slutsky Equation?

It is the only way of knowing how big income and

substitution effects are.

Combined with elasticity estimates it tells us that income

effects are too small to bother with except for goods that

are a large proportion of the budget.

0 x1

x2

gradient – p1/p2

gradient – (p1 + p1)/p2A

B

C

A to B, change in compensated demand. This is the

substitution effect.

B to C income effect. This is the shift in uncompensated

demand when income m changes.

h1(p1, p2,u) compensated demand for good 1.

x 1(p1, p2,m) uncompensated demand for good 1.

Intuition for the Slutsky Equation

When p1 rises to p1 + Δp1 this is as if income falls by x1 Δp1

because this is how much more it costs to buy x1, so the

effective change in income is Δm = -x1 Δp1 .

The change in x1 through the income effect is approximately

The change in x1 through the substitution effect is approximately

Total change

so

1111 px

m

xm

m

x

1

1

1 pp

h

111

1

1

11 px

m

xp

p

hx

11

1

1

1

1 xm

x

p

h

p

x

m

xp

m

x

x

m

p

h

h

p

p

x

x

p

hxhpxp

xm

x

p

h

p

x

mppxm

mppx

p

uxxh

p

mppx

uppEm

111

11

1

1

1

1

1

1

1

111111

11

1

1

1

1

211211

1

211

1

211

21

because //by multiply

down writeeasier to thismake toarguments out the leave

),,(),,(),,(),,(

),,(When

Slutsky equation in elasticities

Slutsky equation

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1

But what are these elasticities?

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1

Own price

elasticity of

uncompensated

demand

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1

Own price

elasticity of

compensated

demand

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1

Income elasticity

of

uncompensated

demand

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1

budget

share

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1

Own price elasticity

of uncompensated

demand

Own price

elasticity of

compensated

demand

Income

elasticity of

uncompensated

demand

budget

share

*

Perloff notation

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1

Income effects and the Slutsky Equation

Sign? Sign for a normal

good

Sign for an inferior

good

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1


Sign?

negative

Sign for a normal

good


good

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1


Sign?

negative

Sign for a normal

good positive


good

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1


Sign?

negative

Sign for a normal

good positive


good negative.

m

xp

m

x

x

m

p

h

h

p

p

x

x

p 111

11

1

1

1

1

1

1

1


budget shareincome elasticity

Income effects are small for goods with a small budget

share, even if the income elasticity of demand is quite

large.

Quantitatively income effects often don’t matter.

Quantity of good x1

Price of

good

p1

Elasticity of demand curvesA

B

Which demand curve is more elastic A or B?

If good 1 is a normal good which is more elastic,

compensated or uncompensated demand?

If good 1 is an inferior good which is more elastic,


Quantity of good x1

Price of

good

p1


B





compensated or uncompensated demand? TP

Quantity of good x1

Price of

good

p1


B






Quantity of good x1

Price of

good

p1


B






Expenditure minimization & compensated

demand. What have we achieved?

• Income and substitution effects. Will be important for

• labour supply, saving & borrowing.

• Downward sloping compensated demand

• Result from the Slutsky equation, income effect is small

if budget share is small.

• Foundation for understanding welfare measures

• consumer surplus, price indices, compensating and

equivalent variation.

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Economics Lecture 4 - Univerzita Karlovavitali/documentsCourses/Lecture 4...Economics Lecture 4 2016-17 Sebastiano Vitali Course Outline 1 Consumer theory and its applications 1.1